\(b\in (0,3]\) , which for \(b=2\)  and \(b=3\)  reduces to the Camassa-Holm and Degasperis-Procesi equations, respectively, see [1]}. It can be rewritten as
\(u_t+uu_x=-\partial _x\Lambda ^{-2}\Big (b̑{2}u^2+{2}u_x^2\Big ).\) 
